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At most 27 length inequalities define Maskit's fundamental domain for the modular group in genus 2

### David Griffiths

Geometry & Topology Monographs 1 (1998) 167–180
DOI: 10.2140/gtm.1998.1.167
 arXiv: math.GT/9811180
##### Abstract
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In recently published work Maskit constructs a fundamental domain ${\mathsc{D}}_{g}$ for the Teichmüller modular group of a closed surface $\mathsc{S}$ of genus $g\ge 2$. Maskit’s technique is to demand that a certain set of $2g$ non-dividing geodesics ${\mathsc{C}}_{2g}$ on $\mathsc{S}$ satisfies certain shortness criteria. This gives an a priori infinite set of length inequalities that the geodesics in ${\mathsc{C}}_{2g}$ must satisfy. Maskit shows that this set of inequalities is finite and that for genus $g=2$ there are at most 45. In this paper we improve this number to 27. Each of these inequalities: compares distances between Weierstrass points in the fundamental domain $\mathsc{S}\setminus {\mathsc{C}}_{4}$ for $\mathsc{S}$; and is realised (as an equality) on one or other of two special surfaces.

##### Keywords
fundamental domain, non-dividing geodesic, Teichmüller modular group, hyperelliptic involution, Weierstrass point
##### Mathematical Subject Classification
Secondary: 14H55, 30F60